11 Probability Applied — Sensor Noise from First Principles
The previous chapter laid out the generic measurement chain. This page works it through end-to-end for one concrete sensor — the camera — because every link corresponds to physically distinct hardware, making it the cleanest worked example. The same template applies to vibration, audio, network telemetry; the table at the bottom shows where each one plugs in.
11.1 1. Random variables and distributions — recap
A random variable is a function that maps an experiment’s outcome to a number. Every sensor reading is a random variable — it takes a different value each time you measure the same underlying state, because the physics of measurement is inherently random.
Three quantities characterise a distribution:
- Mean \mu = E[X] — the expected value
- Variance \sigma^2 = E[(X - \mu)^2] — the spread
- Standard deviation \sigma — same units as X
11.2 2. The distribution chain — worked for the camera
11.2.1 Bernoulli — one photon, one photosite
Each photon either reaches the photosite or it doesn’t. Probability p of detection (= quantum efficiency, QE):
P(X = 1) = p, \qquad P(X = 0) = 1 - p
Mean = p, variance = p(1 - p).
Generic version: one event, two outcomes, probability p of success.
11.2.2 Binomial — n photons, one photosite
Count the number detected in n independent trials:
P(X = k) \;=\; \binom{n}{k} p^k (1 - p)^{n - k}
Mean = np, variance = np(1 - p).
Generic version: successes in n identical independent trials.
11.2.3 Poisson — photon flux model
When n \to \infty and p \to 0 such that np = \lambda stays fixed:
P(X = k) \;=\; \frac{\lambda^k \, e^{-\lambda}}{k!}
Mean = \lambda, variance = \lambda. This is the exact model for photon arrivals over an exposure.
Generic version: rare events arriving at rate \lambda per window — photons per exposure, shock pulses per second, packet drops per minute.
11.2.4 Normal — large-count limit
When \lambda \gg 1, the Poisson distribution converges to a Gaussian:
\text{Poisson}(\lambda) \;\xrightarrow{\lambda \to \infty}\; \mathcal{N}(\mu = \lambda, \sigma^2 = \lambda)
This is why shot noise is Gaussian in bright conditions.
11.2.5 Central Limit Theorem — why sums are always Gaussian
In a sensor, multiple noise sources add together (shot + read + dark + quantization). Even if each source is non-Gaussian, the CLT says the sum is approximately Gaussian. This is why total sensor noise is well-modelled as Gaussian in almost all practical cases — and the same conclusion holds for a vibration sensor, an audio ADC, or any other measurement front-end.
11.3 3. Shot noise is irreducible
From the Poisson model: \sigma_{\text{shot}} = \sqrt\lambda.
This cannot be reduced by better hardware — it is a fundamental property of the discrete nature of light. To halve the shot noise you must quadruple the photon count (larger aperture, longer exposure, bigger photosite).
\text{SNR} \;=\; \frac{\lambda}{\sqrt\lambda} \;=\; \sqrt\lambda
Doubling the photon count improves SNR by \sqrt 2 \approx 41\,\%. The same \sigma = \sqrt\lambda rule holds for any Poisson process: bearing-fault counts, packet drops, Geiger-counter clicks.
11.4 4. Noise budget
A real sensor has multiple noise sources. The total noise variance is their sum (assuming independence):
\sigma_{\text{total}}^2 \;=\; \sigma_{\text{shot}}^2 + \sigma_{\text{read}}^2 + \sigma_{\text{dark}}^2 + \sigma_{\text{quant}}^2 \;=\; \lambda + \sigma_r^2 + \sigma_d^2 + \frac{\Delta^2}{12}
where \Delta is the quantization step size.
Which term dominates depends on the signal regime:
| Regime | Dominant noise | Characteristic |
|---|---|---|
| Bright (\lambda \gg \sigma_r^2) | Shot noise | \sigma \propto \sqrt\lambda |
| Dim (\lambda \ll \sigma_r^2) | Read noise | \sigma \approx \sigma_r (constant floor) |
| Saturated (\lambda \geq C) | Clipping | All information above C is lost |
In bright conditions, collecting more signal always helps. In dim conditions, electronics dominate and more signal doesn’t help much until you climb out of the electronics floor.
The same regime structure shows up in any measurement chain: an electronics-limited floor at low signal, a physics-limited (often \sqrt\lambda) regime at medium signal, and a saturation regime where the measurement is destroyed.
11.5 5. The Anscombe transform
Poisson noise has variance equal to its mean — awkward statistically. The Anscombe transform converts Poisson noise to approximately unit-variance Gaussian:
A(X) \;=\; 2\sqrt{X + \tfrac{3}{8}}
After applying A, the data has approximately constant variance \approx 1 regardless of \lambda. This lets standard Gaussian-assumption algorithms work correctly on Poisson-distributed data — useful any time you have count data you want to feed into a tool that assumes constant-variance Gaussian noise.
11.6 6. The same chain — where other sensors plug in
| Stage | Camera | Vibration (accelerometer) | Network telemetry | Audio (microphone) |
|---|---|---|---|---|
| Bernoulli event | Photon detected (QE) | Shock pulse in a time slice | Packet dropped or not | Pressure crossing threshold |
| Binomial count | Photons detected per exposure | Pulses per window | Drops per minute | Threshold crossings per frame |
| Poisson regime | \lambda = expected photons | \lambda = expected pulses/hour | \lambda = expected drops/min | (rarely the dominant model) |
| Gaussian limit | High illumination | Many pulses per window | High-traffic monitoring | Almost always — the noise floor is Gaussian to begin with |
| Electronics floor | Read noise | Charge-amp + ADC noise | (none) | Pre-amp + ADC noise |
| Quantization | ADC bit depth | ADC bit depth | (count, no quantization) | 16/24-bit ADC |
The chain is the same. Only the names change.
11.7 Summary
| Concept | Key fact |
|---|---|
| Bernoulli | One trial, two outcomes |
| Binomial | n trials, count of successes |
| Poisson | n \to \infty, p \to 0, np = \lambda; mean = variance = \lambda |
| Gaussian | Limiting case \lambda \to \infty — why most noise floors are approximately Normal |
| CLT | Sum of any independent RVs → Gaussian; why total sensor noise is Normal |
| SNR | = \sqrt\lambda in any Poisson regime (photons, pulses, packets) |
| Noise budget | \sigma_{\text{total}}^2 = \sigma_{\text{physics}}^2 + \sigma_{\text{electronics}}^2 + \sigma_{\text{quant}}^2 |